# Loads data
load("bc_bear_occurrences.Rda")
load("BC_Covariates.Rda")
summary(DATA)
##            Length Class           Mode
## Window      1     SpatialPolygons S4  
## Elevation  10     im              list
## Forest     10     im              list
## HFI        10     im              list
## Dist_Water 10     im              list

Visualize bear occurrences in BC

# Extracts location columns
bears_loc <- occ_data[, c("decimalLongitude", "decimalLatitude", "month", "year")]
bears_loc_filtered <- subset(bears_loc, year %in% c(2020, 2021, 2022, 2023, 2024))

# Creates sf object
bears_sf <- st_as_sf(
  bears_loc_filtered,
  coords = c("decimalLongitude", "decimalLatitude"),
  crs = 4326  # WGS84 (longitude/latitude)
)

# BC Albers projection
bears_sf_proj <- st_transform(bears_sf, crs = 3005)

# Extracts coordinates
coords <- st_coordinates(bears_sf_proj)

# Extracts BC window
window_sf <- st_as_sf(DATA$Window) 
window_proj <- st_transform(window_sf, crs = 3005) 
window <- as.owin(window_proj) 

# Creates ppp object
bears_ppp <- ppp(
  x = coords[,1],
  y = coords[,2],
  window = window
)
## Warning: 187 points were rejected as lying outside the specified window
## Warning: data contain duplicated points
plot(bears_ppp, pch = 21, main = "Black bear occurrences in BC, 2024")
## Warning in plot.ppp(bears_ppp, pch = 21, main = "Black bear occurrences in BC,
## 2024"): 187 illegal points also plotted

Compare seasonal distributions

# Defines the seasons and corresponding colors
seasons <- list(
  winter = c(12, 1, 2),
  spring = c(3, 4, 5),
  summer = c(6, 7, 8),
  autumn = c(9, 10, 11)
)

# Create an empty list to store the ppp objects for each season
ppp_list <- list()

# Create an empty list to store the filtered data for each season
bears_sf_list <- list()

# Loop over seasons
for (i in 1:length(seasons)) {
  
  # Filter for each season
  season_name <- names(seasons)[i]
  season_months <- seasons[[i]]
  
  # Filter bears_loc_filtered by the season's months
  bears_loc_season <- bears_loc_filtered[bears_loc_filtered$month %in% season_months, ]
  
  # Print the number of observations for this season
  cat(season_name, ": ", nrow(bears_loc_season), " observations\n", sep="")
  
  # Create sf object for the filtered season
  bears_sf_season <- st_as_sf(
    bears_loc_season,
    coords = c("decimalLongitude", "decimalLatitude"),
    crs = 4326  # WGS84 (longitude/latitude)
  )
  
  # Transform to BC Albers projection
  bears_sf_season_proj <- st_transform(bears_sf_season, crs = 3005)
  
  # Store sf object in the list
  bears_sf_list[[season_name]] <- bears_sf_season_proj
  
  # Extract x, y coordinates
  coords <- st_coordinates(bears_sf_season_proj)
  
  # Create ppp object for each season
  suppressWarnings(
    bears_ppp_season <- ppp(
      x = coords[, 1],
      y = coords[, 2],
      window = window
    )
  )
  
  # Store ppp object in the list
  ppp_list[[season_name]] <- bears_ppp_season
}
## winter: 76 observations
## spring: 808 observations
## summer: 1862 observations
## autumn: 841 observations
# Plot the four maps in a 2x2 grid
par(mfrow=c(2,2), mar=c(1,1,1,1))  # Set up a 2x2 plotting window

# Loop over ppp objects and plot them
for (i in 1:length(ppp_list)) {
  season_name <- names(ppp_list)[i]
  suppressWarnings(
    plot(ppp_list[[season_name]], 
         main = paste(season_name, "(2024)"), 
         pch = 21)
  )
}

plot(DATA$Elevation)

plot(DATA$Forest)

plot(DATA$HFI)

plot(DATA$Dist_Water)

elev <- DATA$Elevation
cover <- DATA$Forest
dist_water <- DATA$Dist_Water
hfi <- DATA$HFI

Estimating the intensity of bear occurrences based on elevation

# intensity as a function of elev
par(mfrow=c(2,2), mar=c(2,2,2,2))
for (i in 1:length(ppp_list)) {
  season_name <- names(ppp_list)[i]
  rho <- rhohat(ppp_list[[season_name]], elev) 
  plot(rho, xlim=c(0, max(elev)), main = paste(season_name))
}

# Loads elevation raster
elev <- DATA$Elevation

# Converts im to raster
im2raster <- function(im_obj) {
  m <- as.matrix(im_obj)
  r <- raster(m)
  ext <- c(im_obj$xrange[1] - im_obj$xstep/2,
           im_obj$xrange[2] + im_obj$xstep/2,
           im_obj$yrange[1] - im_obj$ystep/2,
           im_obj$yrange[2] + im_obj$ystep/2)
  extent(r) <- ext
  return(r)
}
elev_raster <- im2raster(elev)

# Extracts elevation values at occurrence locations
bears_sf_proj$elev_value <- extract(elev_raster, bears_sf_proj)
summary(bears_sf_proj$elev_value)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   11.68  461.09  644.88  754.84  937.91 2308.18    1499

KDE to estimate the concentration of bear occurrences based on elevation values. The elevation data for bear locations ranges from a minimum of 11.68 meters to a maximum of 2308.18 meters, with most occurrences concentrated between 461.09 meters and 937.91 meters. The mean elevation is 754.84 meters, slightly higher than the median, indicating a skew towards higher elevations.

# KDE for Bear Locations
kde_b <- density(bears_sf_proj$elev_value, na.rm = TRUE, bw = "SJ-dpi")
summary(bears_sf_proj$elev_value)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   11.68  461.09  644.88  754.84  937.91 2308.18    1499
plot(kde_b,
     main = "Kernel Density Estimate of Elevation at Bear Locations",
     xlab = "Elevation (meters)",
     ylab = "Density",
     col = "#2C6B2F", 
     lwd = 2)  

Fitted model: Intercept: -17.37; Elevation coefficient: -0.0025. The negative coefficient for elevation indicates that, as elevation increases, the expected intensity of bear occurrences decreases. This suggests that bears are less likely to be found at higher elevations.

# NA with mean elevation
mean_elev <- mean(as.vector(as.matrix(elev)), na.rm = TRUE)
elev_clean <- eval.im(ifelse(is.na(elev), mean_elev, elev))

# model
elevation_model <- ppm(bears_ppp, ~ elev, covariates = list(elev = elev_clean))
print(elevation_model)
## Nonstationary Poisson process
## Fitted to point pattern dataset 'bears_ppp'
## 
## Log intensity:  ~elev
## 
## Fitted trend coefficients:
##  (Intercept)         elev 
## -17.36871576  -0.00253243 
## 
##                 Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -17.36871576 2.963479e-02 -17.426798885 -17.310632643   ***
## elev         -0.00253243 4.234993e-05  -0.002615434  -0.002449426   ***
##                   Zval
## (Intercept) -586.09208
## elev         -59.79774
# Predicts and plots intensity
pred_intensity_elev <- predict(elevation_model)
plot(pred_intensity_elev, 
     main = "Predicted Intensity Based on Elevation")
points(bears_ppp, 
       pch = 20, 
       cex = 0.4,
       col = "#B24A2F")

Computing and plotting residuals of the point process model fitted to elevation data. Using Pearson residuals and removing any non-finite residuals

elevation_residuals <- residuals(elevation_model, type = "pearson")
elevation_residuals$v[!is.finite(elevation_residuals$v)] <- NA
plot(elevation_residuals, 
     main = "Residuals - Elevation Model", 
     na.col = "transparent")

Winter: Intercept: -20.28; Elevation Coefficient: -0.0049. For winter, the negative coefficient for elevation indicates that bear occurrences decrease as elevation increases. The model did not converge, so caution is needed in interpreting the results.

Spring: Intercept: -18.94; Elevation Coefficient: -0.0024. In spring, the relationship between elevation and bear occurrences is also negative, with a slight decrease in occurrences as elevation increases. However, the model showed some issues with convergence.

Summer: Intercept: -18.21; Elevation Coefficient: -0.0022. Summer shows a similar negative relationship, with bear occurrences decreasing slightly as elevation increases.

Autumn: Intercept: -18.51; Elevation Coefficient: -0.0031. In autumn, bear occurrences also decrease with elevation, with a slightly stronger negative coefficient compared to spring and summer.

####All seasons show a negative relationship between bear occurrences and elevation. Winter has the largest negative coefficient (-0.0049), meaning a stronger decrease in occurrences with elevation.

# Seasonal elevation models
models_elev_seasonal <- list()
for (season in names(ppp_list)) {
  models_elev_seasonal[[season]] <- ppm(ppp_list[[season]], ~ elev, covariates = list(elev = elev_clean))
  cat("Model for", season, ":\n")
  print(models_elev_seasonal[[season]])
}
## Warning: glm.fit: algorithm did not converge
## Model for winter :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
## 
## Log intensity:  ~elev
## 
## Fitted trend coefficients:
##   (Intercept)          elev 
## -20.281500393  -0.004896402 
## 
##                  Estimate         S.E.       CI95.lo      CI95.hi Ztest
## (Intercept) -20.281500393 0.1590970890 -20.593324958 -19.96967583   ***
## elev         -0.004896402 0.0004162485  -0.005712234  -0.00408057   ***
##                   Zval
## (Intercept) -127.47876
## elev         -11.76317
## *** Fitting algorithm for 'glm' did not converge ***
## Model for spring :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
## 
## Log intensity:  ~elev
## 
## Fitted trend coefficients:
##   (Intercept)          elev 
## -18.938423756  -0.002390713 
## 
##                  Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -18.938423756 6.296883e-02 -19.061840399 -18.815007114   ***
## elev         -0.002390713 8.708531e-05  -0.002561397  -0.002220029   ***
##                   Zval
## (Intercept) -300.75870
## elev         -27.45254
## Model for summer :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
## 
## Log intensity:  ~elev
## 
## Fitted trend coefficients:
##   (Intercept)          elev 
## -18.213839873  -0.002227723 
## 
##                  Estimate         S.E.       CI95.lo      CI95.hi Ztest
## (Intercept) -18.213839873 4.265390e-02 -18.297439975 -18.13023977   ***
## elev         -0.002227723 5.699771e-05  -0.002339437  -0.00211601   ***
##                   Zval
## (Intercept) -427.01468
## elev         -39.08443
## Model for autumn :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
## 
## Log intensity:  ~elev
## 
## Fitted trend coefficients:
##   (Intercept)          elev 
## -18.514156648  -0.003139775 
## 
##                  Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -18.514156648 5.671064e-02 -18.625307457 -18.403005838   ***
## elev         -0.003139775 9.455484e-05  -0.003325099  -0.002954451   ***
##                   Zval
## (Intercept) -326.46708
## elev         -33.20586
# Combines seasonal data
bears_all_elevation <- do.call(rbind, lapply(names(bears_sf_list), function(season) {
  sf_season <- bears_sf_list[[season]]
  coords <- st_coordinates(sf_season)
  elev_vals <- extract(elev_raster, sf_season)
  data.frame(x = coords[, 1], y = coords[, 2],
             elev_value = elev_vals, season = season)
}))
bears_all_elevation$season <- as.factor(bears_all_elevation$season)
bears_all_elevation$count <- 1

GAM results suggest that neither elevation nor season has a significant influence on the spatial distribution of bear locations. Warnings about the model fitting process suggest that the results may not be reliable.

# GAM with elevation and season
gam_elevation <- gam(count ~ s(elev_value) + season, data = bears_all_elevation, family = poisson())
summary(gam_elevation)
## 
## Family: poisson 
## Link function: log 
## 
## Formula:
## count ~ s(elev_value) + season
## 
## Parametric coefficients:
##               Estimate Std. Error z value Pr(>|z|)
## (Intercept)  6.729e-17  4.510e-02       0        1
## seasonspring 2.172e-22  6.531e-02       0        1
## seasonsummer 7.375e-22  5.431e-02       0        1
## seasonwinter 8.153e-21  1.544e-01       0        1
## 
## Approximate significance of smooth terms:
##               edf Ref.df Chi.sq p-value
## s(elev_value)   1  1.001      0       1
## 
## R-sq.(adj) =    NaN   Deviance explained =  NaN%
## UBRE = -0.99521  Scale est. = 1         n = 2088

Model shows a negative relationship between bear locations and elevation, suggesting bears are less likely to be found at higher elevations. Quadratic term suggests a slight non-linear effect, but primary relationship appears negative. Model also indicates that summer sees the highest concentration of bear points, while winter has an expected decrease in bear points. Spring also shows a negative effect, although it is smaller than winter’s effect.

Bears tend to prefer lower elevations (based on the negative relationship with elevation). The summer season has the highest bear presence, while winter and spring have much lower bear occurrences, as to be expected.

# Polynomial ppm model with season
bears_all_elevation <- ppp(
  x = bears_all_elevation$x,
  y = bears_all_elevation$y,
  window = window,
  marks = bears_all_elevation$season
)
## Warning: 187 points were rejected as lying outside the specified window
## Warning: data contain duplicated points
marks(bears_all_elevation) <- data.frame(season = marks(bears_all_elevation))

model_poly_elevation <- ppm(bears_all_elevation, ~ polynom(elev, 2) + marks,
                     covariates = list(elev = elev_clean))
print(model_poly_elevation)
## Nonstationary multitype Poisson process
## Fitted to point pattern dataset 'bears_all_elevation'
## 
## Possible marks: 'autumn', 'spring', 'summer' and 'winter'
## 
## Log intensity:  ~elev + I(elev^2) + marks
## 
## Fitted trend coefficients:
##   (Intercept)          elev     I(elev^2)   marksspring   markssummer 
## -1.850394e+01 -3.937116e-03  9.510112e-07 -3.957121e-02  7.891398e-01 
##   markswinter 
## -2.379296e+00 
## 
##                  Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -1.850394e+01 4.652449e-02 -1.859513e+01 -1.841276e+01   ***
## elev        -3.937116e-03 1.034097e-04 -4.139795e-03 -3.734437e-03   ***
## I(elev^2)    9.510112e-07 6.143974e-08  8.305915e-07  1.071431e-06   ***
## marksspring -3.957121e-02 5.053363e-02 -1.386153e-01  5.947288e-02      
## markssummer  7.891398e-01 4.266227e-02  7.055233e-01  8.727563e-01   ***
## markswinter -2.379296e+00 1.215116e-01 -2.617454e+00 -2.141137e+00   ***
##                     Zval
## (Intercept) -397.7247657
## elev         -38.0729907
## I(elev^2)     15.4787625
## marksspring   -0.7830669
## markssummer   18.4973701
## markswinter  -19.5808065
# Compares models using AIC
cat("AIC for poly model: ", AIC(model_poly_elevation), "\n")
## AIC for poly model:  142038.1
cat("AIC for gam model: ", AIC(gam_elevation), "\n")
## AIC for gam model:  4186.001

Creating a point pattern object for each season

# Creates seasonal ppp objects
ppp_list <- list()
for (s in names(seasons)) {
  bears_loc_season <- bears_loc_filtered[bears_loc_filtered$month %in% seasons[[s]], ]
  bears_sf_season <- st_as_sf(bears_loc_season, coords = c("decimalLongitude", "decimalLatitude"), crs = 4326)
  bears_sf_season_proj <- st_transform(bears_sf_season, crs = 3005)
  coords_season <- st_coordinates(bears_sf_season_proj)
  suppressWarnings(
    ppp_list[[s]] <- ppp(x = coords_season[,1], y = coords_season[,2], window = window)
  )
}

# Extracts covariates
elevation <- DATA$Elevation
cover <- DATA$Forest
hfi <- DATA$HFI
dist_water <- DATA$Dist_Water

Calculating and plotting the KDE for bear occurrences for each season

# KDE Surface Comparison
dens_list <- lapply(ppp_list, function(p) density(p, sigma = 10000))
for (s in names(dens_list)) {
  plot(dens_list[[s]], main = paste("KDE Surface -", s))
}

Performing Quadrat test for each season to analyze the spatial pattern of bear occurrences. Dividing area into a grid of 5x5 and testing for deviations from a random distribution. Bear locations do not follow a random distribution and show spatial clustering across all seasons, meaning their presence is likely influenced by specific environmental or ecological factors that are season-dependent. The quadrant test results suggest that for all seasons (winter, spring, summer, and autumn), bear locations exhibit significant spatial clustering or non-random patterns rather than being randomly distributed (as indicated by the very small p-values).

Bear distribution is influenced by factors other than randomness, such as environmental features, resource availability, or behavior patterns that vary across seasons.

Winter: X2 = 561.39, p-value < 2.2e-16. P-value indicates a strong deviation from CSR for winter, suggesting that bear locations are not randomly distributed in the winter season.

Spring: X2 = 2202.7, p-value < 2.2e-16. Test statistic is very large, and the p-value is highly significant, indicating that the bear locations in spring also show non-random clustering or patterns.

Summer: X2 = 3767, p-value < 2.2e-16. Summer further supports a non-random pattern, with an even higher test statistic and significant p-value.

Autumn: X2 = 2720.5, p-value < 2.2e-16. Autumn shows a significant departure from CSR, confirming that bear locations are non-randomly distributed.

# quadrat tests
for (s in names(ppp_list)) {
  qt <- quadrat.test(ppp_list[[s]], nx = 5, ny = 5)
  print(qt)
  plot(qt, main = paste("Quadrat test -", s))
}
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
## 
##  Chi-squared test of CSR using quadrat counts
## 
## data:  ppp_list[[s]]
## X2 = 561.39, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
## 
## Quadrats: 21 tiles (irregular windows)

## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
## 
##  Chi-squared test of CSR using quadrat counts
## 
## data:  ppp_list[[s]]
## X2 = 2202.7, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
## 
## Quadrats: 21 tiles (irregular windows)

## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
## 
##  Chi-squared test of CSR using quadrat counts
## 
## data:  ppp_list[[s]]
## X2 = 3767, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
## 
## Quadrats: 21 tiles (irregular windows)

## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
## 
##  Chi-squared test of CSR using quadrat counts
## 
## data:  ppp_list[[s]]
## X2 = 2720.5, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
## 
## Quadrats: 21 tiles (irregular windows)

Fitting an inhomogeneous Poisson ppm for each season, using elevation. Model used to predict intensity and plotted for each season.

# inhomogeneous poission model
for (s in names(ppp_list)) {
  model <- ppm(ppp_list[[s]] ~ elev, covariates = list(elev = elevation))
  plot(predict(model), main = paste("Intensity ~ Elevation -", s))
}
## Warning: glm.fit: algorithm did not converge

## Warning: Values of the covariate 'elev' were NA or undefined at 0.02% (1 out of
## 5456) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev, data = NULL, interaction = NULL,

Fitting a multivariate Poisson Point Process model to bear occurrences for each season, considering all covariates: elevation, forest cover, hfi, and distance to water. Predicted intensity plotted for each season.

# multivariate ppm
for (s in names(ppp_list)) {
  model <- ppm(ppp_list[[s]] ~ elev + cover + hfi + dist_water,
               covariates = list(elev = elevation, cover = cover, hfi = hfi, dist_water = dist_water))
  print(summary(model))
  plot(predict(model), main = paste("Predicted Intensity -", s))
}
## Warning: Values of the covariate 'hfi' were NA or undefined at 0.68% (4 out of
## 587) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,
## Warning: glm.fit: algorithm did not converge
## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## *** Algorithm did not converge ***
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water, 
##     covariates = list(elev = elevation, cover = cover, hfi = hfi, 
##         dist_water = dist_water))
## Edge correction: "border"
##  [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
## 
## Data pattern:
## Planar point pattern:  74 points
## Average intensity 7.8e-11 points per square unit
## Window: polygonal boundary
## 378 separate polygons (2 holes)
##                    vertices         area relative.area
## polygon 1              5454 889865000000      9.38e-01
## polygon 2                19     11135700      1.17e-05
## polygon 3                 6      1679950      1.77e-06
## polygon 4                59    375017000      3.95e-04
## polygon 5                25     54751800      5.77e-05
## polygon 6                 5      1220870      1.29e-06
## polygon 7                 8      6790020      7.16e-06
## polygon 8                63    151745000      1.60e-04
## polygon 9                12     10733900      1.13e-05
## polygon 10                3       732692      7.73e-07
## polygon 11               10      6152880      6.49e-06
## polygon 12                4       808047      8.52e-07
## polygon 13               15     12418500      1.31e-05
## polygon 14                7      2619570      2.76e-06
## polygon 15               24     19177800      2.02e-05
## polygon 16                4       637359      6.72e-07
## polygon 17                7      1083120      1.14e-06
## polygon 18                4       630075      6.64e-07
## polygon 19               16     23094400      2.44e-05
## polygon 20                4       740351      7.81e-07
## polygon 21               28     31469400      3.32e-05
## polygon 22                5      1243830      1.31e-06
## polygon 23               25     33536100      3.54e-05
## polygon 24               30     53162300      5.61e-05
## polygon 25              751   6476610000      6.83e-03
## polygon 26               34     67360500      7.10e-05
## polygon 27                5      2010320      2.12e-06
## polygon 28               21     50455600      5.32e-05
## polygon 29                8      2267800      2.39e-06
## polygon 30                6      2616700      2.76e-06
## polygon 31               13      9776920      1.03e-05
## polygon 32               14     16276800      1.72e-05
## polygon 33                3       567574      5.99e-07
## polygon 34                4      1110880      1.17e-06
## polygon 35              189    531172000      5.60e-04
## polygon 36                7      3190600      3.36e-06
## polygon 37                9      4705250      4.96e-06
## polygon 38               18     44300200      4.67e-05
## polygon 39                9      3016490      3.18e-06
## polygon 40                4       846360      8.93e-07
## polygon 41               10      6016930      6.35e-06
## polygon 42                5      1647440      1.74e-06
## polygon 43                9      2913680      3.07e-06
## polygon 44                5      1104420      1.16e-06
## polygon 45               14     14571400      1.54e-05
## polygon 46                5      1978280      2.09e-06
## polygon 47              281   1397560000      1.47e-03
## polygon 48               20     35892800      3.79e-05
## polygon 49                5      1193280      1.26e-06
## polygon 50               21     34874700      3.68e-05
## polygon 51               11     10881600      1.15e-05
## polygon 52                4       501226      5.29e-07
## polygon 53                5      1071650      1.13e-06
## polygon 54                3       429629      4.53e-07
## polygon 55               17     16117900      1.70e-05
## polygon 56               72    284295000      3.00e-04
## polygon 57               15     11766100      1.24e-05
## polygon 58                5      2151150      2.27e-06
## polygon 59                7      2258780      2.38e-06
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##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## 
## Dummy quadrature points:
##      32 x 32 grid of dummy points, plus 4 corner points
##      dummy spacing: 49896.83 x 42706.99 units
## 
## Original dummy parameters: =
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## Average intensity 5.41e-10 points per square unit
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## enclosing rectangle: [273874.9, 1870573.4] x [369042.8, 1735666.4] units
##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## Quadrature weights:
##      (counting weights based on 32 x 32 array of rectangular tiles)
## All weights:
##  range: [55500000, 2.13e+09] total: 9.47e+11
## Weights on data points:
##  range: [59300000, 1.07e+09] total: 2.75e+10
## Weights on dummy points:
##  range: [55500000, 2.13e+09] total: 9.2e+11
## --------------------------------------------------------------------------------
## FITTED :
## 
## Nonstationary Poisson process
## 
## ---- Intensity: ----
## 
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
##  elev: im
##  cover: im
##  hfi: im
##  dist_water: im
## 
## Fitted trend coefficients:
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.345970e+01 -3.511481e-03  1.690825e-02  4.869412e+00  9.553740e-05 
## 
##                  Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -2.345970e+01 5.287726e-01 -2.449607e+01 -22.423323608   ***
## elev        -3.511481e-03 4.743760e-04 -4.441241e-03  -0.002581722   ***
## cover        1.690825e-02 5.095351e-03  6.921542e-03   0.026894951   ***
## hfi          4.869412e+00 5.529194e-01  3.785710e+00   5.953114352   ***
## dist_water   9.553740e-05 5.890243e-05 -1.990924e-05   0.000210984      
##                   Zval
## (Intercept) -44.366332
## elev         -7.402317
## cover         3.318367
## hfi           8.806731
## dist_water    1.621960
## 
## ----------- gory details -----
## 
## Fitted regular parameters (theta):
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.345970e+01 -3.511481e-03  1.690825e-02  4.869412e+00  9.553740e-05 
## 
## Fitted exp(theta):
##  (Intercept)         elev        cover          hfi   dist_water 
## 6.480108e-11 9.964947e-01 1.017052e+00 1.302443e+02 1.000096e+00 
## Problem:
##  Values of the covariate 'hfi' were NA or undefined at 0.68% (4 out of 587) of the quadrature points

## Warning: Values of the covariate 'hfi' were NA or undefined at 0.45% (12 out of
## 2667) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,
## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## Algorithm converged
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water, 
##     covariates = list(elev = elevation, cover = cover, hfi = hfi, 
##         dist_water = dist_water))
## Edge correction: "border"
##  [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
## 
## Data pattern:
## Planar point pattern:  768 points
## Average intensity 8.1e-10 points per square unit
## Window: polygonal boundary
## 378 separate polygons (2 holes)
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##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## 
## Dummy quadrature points:
##      64 x 64 grid of dummy points, plus 4 corner points
##      dummy spacing: 24948.41 x 21353.49 units
## 
## Original dummy parameters: =
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## Average intensity 2e-09 points per square unit
## Window: polygonal boundary
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## enclosing rectangle: [273874.9, 1870573.4] x [369042.8, 1735666.4] units
##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## Quadrature weights:
##      (counting weights based on 64 x 64 array of rectangular tiles)
## All weights:
##  range: [6750000, 5.33e+08]  total: 9.45e+11
## Weights on data points:
##  range: [6750000, 2.92e+08]  total: 6.97e+10
## Weights on dummy points:
##  range: [6750000, 5.33e+08]  total: 8.75e+11
## --------------------------------------------------------------------------------
## FITTED :
## 
## Nonstationary Poisson process
## 
## ---- Intensity: ----
## 
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
##  elev: im
##  cover: im
##  hfi: im
##  dist_water: im
## 
## Fitted trend coefficients:
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.111907e+01 -1.478995e-03  9.975132e-03  4.838300e+00 -3.936119e-05 
## 
##                  Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -2.111907e+01 1.352834e-01 -2.138422e+01 -2.085392e+01   ***
## elev        -1.478995e-03 9.298462e-05 -1.661242e-03 -1.296749e-03   ***
## cover        9.975132e-03 1.304182e-03  7.418982e-03  1.253128e-02   ***
## hfi          4.838300e+00 1.561870e-01  4.532179e+00  5.144421e+00   ***
## dist_water  -3.936119e-05 2.095225e-05 -8.042685e-05  1.704473e-06      
##                    Zval
## (Intercept) -156.109836
## elev         -15.905807
## cover          7.648573
## hfi           30.977604
## dist_water    -1.878614
## 
## ----------- gory details -----
## 
## Fitted regular parameters (theta):
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.111907e+01 -1.478995e-03  9.975132e-03  4.838300e+00 -3.936119e-05 
## 
## Fitted exp(theta):
##  (Intercept)         elev        cover          hfi   dist_water 
## 6.731366e-10 9.985221e-01 1.010025e+00 1.262546e+02 9.999606e-01 
## Problem:
##  Values of the covariate 'hfi' were NA or undefined at 0.45% (12 out of 2667) of the quadrature points

## Warning: Values of the covariates 'elev', 'hfi' were NA or undefined at 0.4%
## (22 out of 5456) of the quadrature points. Occurred while executing: ppm.ppp(Q
## = ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,
## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## Algorithm converged
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water, 
##     covariates = list(elev = elevation, cover = cover, hfi = hfi, 
##         dist_water = dist_water))
## Edge correction: "border"
##  [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
## 
## Data pattern:
## Planar point pattern:  1759 points
## Average intensity 1.85e-09 points per square unit
## Window: polygonal boundary
## 378 separate polygons (2 holes)
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##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## 
## Dummy quadrature points:
##      90 x 90 grid of dummy points, plus 4 corner points
##      dummy spacing: 17741.09 x 15184.71 units
## 
## Original dummy parameters: =
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## Average intensity 3.9e-09 points per square unit
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## enclosing rectangle: [273874.9, 1870573.4] x [369042.8, 1735666.4] units
##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## Quadrature weights:
##      (counting weights based on 90 x 90 array of rectangular tiles)
## All weights:
##  range: [2760000, 2.69e+08]  total: 9.45e+11
## Weights on data points:
##  range: [2760000, 1.35e+08]  total: 8.26e+10
## Weights on dummy points:
##  range: [2760000, 2.69e+08]  total: 8.63e+11
## --------------------------------------------------------------------------------
## FITTED :
## 
## Nonstationary Poisson process
## 
## ---- Intensity: ----
## 
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
##  elev: im
##  cover: im
##  hfi: im
##  dist_water: im
## 
## Fitted trend coefficients:
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.046112e+01 -1.310042e-03  1.045653e-02  4.990537e+00 -3.885610e-05 
## 
##                  Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -2.046112e+01 9.180183e-02 -2.064105e+01 -2.028120e+01   ***
## elev        -1.310042e-03 6.134897e-05 -1.430284e-03 -1.189800e-03   ***
## cover        1.045653e-02 8.686491e-04  8.754013e-03  1.215906e-02   ***
## hfi          4.990537e+00 1.087027e-01  4.777484e+00  5.203591e+00   ***
## dist_water  -3.885610e-05 1.341862e-05 -6.515611e-05 -1.255610e-05    **
##                    Zval
## (Intercept) -222.883618
## elev         -21.353940
## cover         12.037696
## hfi           45.909988
## dist_water    -2.895686
## 
## ----------- gory details -----
## 
## Fitted regular parameters (theta):
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.046112e+01 -1.310042e-03  1.045653e-02  4.990537e+00 -3.885610e-05 
## 
## Fitted exp(theta):
##  (Intercept)         elev        cover          hfi   dist_water 
## 1.299710e-09 9.986908e-01 1.010511e+00 1.470154e+02 9.999611e-01 
## Problem:
##  Values of the covariates 'elev', 'hfi' were NA or undefined at 0.4% (22 out of 5456) of the quadrature points

## Warning: Values of the covariate 'hfi' were NA or undefined at 0.67% (18 out of
## 2698) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,
## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## Algorithm converged
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water, 
##     covariates = list(elev = elevation, cover = cover, hfi = hfi, 
##         dist_water = dist_water))
## Edge correction: "border"
##  [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
## 
## Data pattern:
## Planar point pattern:  799 points
## Average intensity 8.43e-10 points per square unit
## Window: polygonal boundary
## 378 separate polygons (2 holes)
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##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## 
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##      dummy spacing: 24948.41 x 21353.49 units
## 
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## polygon 324              27     50295800      5.30e-05
## polygon 325               9      5781470      6.10e-06
## polygon 326              11      7326980      7.73e-06
## polygon 327               5      1123680      1.18e-06
## polygon 328              63    152920000      1.61e-04
## polygon 329               5      2014070      2.12e-06
## polygon 330              64     83219400      8.78e-05
## polygon 331              26     53043500      5.59e-05
## polygon 332               6      3101410      3.27e-06
## polygon 333              26     30659000      3.23e-05
## polygon 334               5       714352      7.53e-07
## polygon 335               4       638063      6.73e-07
## polygon 336               6      1829730      1.93e-06
## polygon 337              23     24218600      2.55e-05
## polygon 338               5      1418840      1.50e-06
## polygon 339               6      1293100      1.36e-06
## polygon 340               5      1691180      1.78e-06
## polygon 341               6      1792550      1.89e-06
## polygon 342               5       885457      9.34e-07
## polygon 343              13     10674100      1.13e-05
## polygon 344              36     56979200      6.01e-05
## polygon 345               3       397429      4.19e-07
## polygon 346              12      9159420      9.66e-06
## polygon 347               4       638721      6.74e-07
## polygon 348               3       374163      3.95e-07
## polygon 349               4       582135      6.14e-07
## polygon 350              10      6826030      7.20e-06
## polygon 351              72    186213000      1.96e-04
## polygon 352               3       772725      8.15e-07
## polygon 353              18     16809400      1.77e-05
## polygon 354               4       617265      6.51e-07
## polygon 355               6      1317820      1.39e-06
## polygon 356               4       545022      5.75e-07
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## polygon 359              23     22590100      2.38e-05
## polygon 360              10      2935450      3.10e-06
## polygon 361               7      2335470      2.46e-06
## polygon 362              13      6217600      6.56e-06
## polygon 363               6      1623490      1.71e-06
## polygon 364               3       422197      4.45e-07
## polygon 365              27     27289800      2.88e-05
## polygon 366               5      1654240      1.74e-06
## polygon 367              20     31802600      3.35e-05
## polygon 368               7      1180430      1.24e-06
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## polygon 370              13      9067690      9.56e-06
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## polygon 374               5      1202270      1.27e-06
## polygon 375              12      7950440      8.38e-06
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## enclosing rectangle: [273874.9, 1870573.4] x [369042.8, 1735666.4] units
##                      (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## Quadrature weights:
##      (counting weights based on 64 x 64 array of rectangular tiles)
## All weights:
##  range: [5180000, 5.33e+08]  total: 9.45e+11
## Weights on data points:
##  range: [5180000, 2.66e+08]  total: 6.3e+10
## Weights on dummy points:
##  range: [7240000, 5.33e+08]  total: 8.82e+11
## --------------------------------------------------------------------------------
## FITTED :
## 
## Nonstationary Poisson process
## 
## ---- Intensity: ----
## 
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
##  elev: im
##  cover: im
##  hfi: im
##  dist_water: im
## 
## Fitted trend coefficients:
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.059377e+01 -2.172077e-03  8.193666e-03  4.581051e+00 -6.675436e-05 
## 
##                  Estimate         S.E.       CI95.lo       CI95.hi Ztest
## (Intercept) -2.059377e+01 1.402255e-01 -2.086860e+01 -2.031893e+01   ***
## elev        -2.172077e-03 1.008390e-04 -2.369717e-03 -1.974436e-03   ***
## cover        8.193666e-03 1.428951e-03  5.392973e-03  1.099436e-02   ***
## hfi          4.581051e+00 1.520399e-01  4.283058e+00  4.879044e+00   ***
## dist_water  -6.675436e-05 2.184644e-05 -1.095726e-04 -2.393613e-05    **
##                    Zval
## (Intercept) -146.861800
## elev         -21.540041
## cover          5.734042
## hfi           30.130590
## dist_water    -3.055617
## 
## ----------- gory details -----
## 
## Fitted regular parameters (theta):
##   (Intercept)          elev         cover           hfi    dist_water 
## -2.059377e+01 -2.172077e-03  8.193666e-03  4.581051e+00 -6.675436e-05 
## 
## Fitted exp(theta):
##  (Intercept)         elev        cover          hfi   dist_water 
## 1.138259e-09 9.978303e-01 1.008227e+00 9.761692e+01 9.999332e-01 
## Problem:
##  Values of the covariate 'hfi' were NA or undefined at 0.67% (18 out of 2698) of the quadrature points

For each season - assessing spatial distribution of bear occurrences (I.e., clustered, dispersed, or randomly distributed)

# Ripley's K with Envelope for each season
for (s in names(ppp_list)) {
  # Compute Ripley's K for the current season
  K <- Kest(ppp_list[[s]])
  E <- envelope(ppp_list[[s]], 
                Kest, 
                correction = "border", 
                rank = 1, 
                nsim = 19, 
                fix.n = TRUE)
  
  plot(K, 
       main = paste("Ripley's K -", s))
  plot(E, 
       add = TRUE,
       lty = 2)  
}
## Generating 19 simulations of CSR with fixed number of points  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 
## 19.
## 
## Done.

## Generating 19 simulations of CSR with fixed number of points  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 
## 19.
## 
## Done.

## Generating 19 simulations of CSR with fixed number of points  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 
## 19.
## 
## Done.

## Generating 19 simulations of CSR with fixed number of points  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 
## 19.
## 
## Done.